Pathway from condensation via fragmentation to fermionization of cold bosonic systems

For small scattering lengths, cold bosonic atoms form a condensate the density profile of which is smooth. With increasing scattering length, the density {\it gradually} acquires more and more oscillations. Finally, the number of oscillations equals the number of bosons and the system becomes {\it fermionized}. On this pathway from condensation to fermionization intriguing phenomena occur, depending on the shape of the trap. These include macroscopic fragmentation and {\it coexistence} of condensed and fermionized parts that are separated in space.